A solid spherical ball and a hollow spherical ball of two different materials of densities `rho_(1) and rho_(2)` respectively have same outer radii and same mass. What will be the ratio of the moment of inertia ( about an axis passing through the centre ) of the hollow sphere to that of the solid sphere?

Options

A. `(rho_(2))/(rho_(1))(1 - (rho_(2))/(rho_(1)))^(5//3)`
B. `(rho_(2))/(rho_(1))[1-(1 - (rho_(2))/(rho_(1)))^(5//3)]`
C. `(rho_(2))/(rho_(1))(1 - (rho_(1))/(rho_(2)))^(5//3)`
D. `(rho_(2))/(rho_(1))[1-(1 - (rho_(1))/(rho_(2)))^(5//3)]`

Answer

D

Explanation

`(4)/(3) pi R^(3) rho_(2) - (4)/(3)pi R_(1)^(3) rho_(2) = (4)/(3) pi R^(3) rho_(1)`
or, `" " R^(3) rho_(2) - R^(3) rho_(1) = R_(1)^(3)rho_(2)`
or, `R_(1) = R(1 - (rho_(1))/(rho_(2)) )^(1//3)`
`therefore (I_("hollow"))/(I_("solid")) = ((4)/(3) pi R^(3) rho_(2) xx (2)/(5)R^(2) - (4)/(3) pi R_(1)^(3) rho_(2) xx (2)/(5) R_(1)^(2))/((4)/(3)pi R^(3) rho_(1) xx (2)/(5)R^(2) )`
= ` (rho_(2) (R^(5) - R_(1)^(5)))/(rho_(1) R^(5)) = (rho_(2))/(rho_(1))(1 - (R_(1)^(5))/(R^(5)))`
` = (rho_(2))/(rho_(1)){ 1 - (1 - (rho_(1))/(rho_(2)))^(5//3) } `